Arabic Mathematics, Indian Mathematics and zero

Al-Khwarizmi

Born: about 780 in Baghdad (now in Iraq)Died: about 850 CE

We know few details of Abu Ja'far Muhammad ibn Musa al-Khwarizmi's life.

perhaps we should call him Al for short One unfortunate effect of this lack of

knowledge seems to be the temptation to make guesses based on very little evidence.

al-Khwarizmi

Having introduced the natural numbers, al-Khwarizmi introduces the main topic of the first section of his book, namely the solution of equations.

His equations are linear or quadratic and are composed of units, roots and squares.

– Known now as the solution by radicals For example, to al-Khwarizmi a unit was a number, a root was

x, and a square was x2. However, although we shall use the now familiar algebraic

notation in this presentation to help us understand the notions, Al-Khwarizmi's mathematics is done entirely in words with no symbols being used.

al-Khwarizmi

He first reduces an equation (linear or quadratic) to one of six standard forms:

1. Squares equal to roots.2. Squares equal to numbers.3. Roots equal to numbers.4. Squares and roots equal to numbers;

e.g. x2 + 10 x = 39.5. Squares and numbers equal to roots;

e.g. x2 + 21 = 10 x.6. Roots and numbers equal to squares;

e.g. 3 x + 4 = x2.

al-Khwarizmi

The reduction is carried out using the two operations of al-jabr and al-muqabala.

Here "al-jabr" means "completion" and is the process of removing negative terms from an equation. It is where we get the word algebra from: Restoration and equivalence

For example, using one of al-Khwarizmi's own examples, "al-jabr" transforms x2 = 40 x - 4 x2 into 5 x2 = 40 x.

The term "al-muqabala" means "balancing" and is the process of reducing positive terms of the same power when they occur on both sides of an equation.

For example, two applications of "al-muqabala" reduces 50 + 3x + x2 = 29 + 10 x to 21 + x2 = 7 x (one application to deal

with the numbers and a second to deal with the roots).

al-Khwarizmi

Al-Khwarizmi then shows how to solve the six standard types of equations.

He uses both algebraic methods of solution and geometric methods. – How can we solve a quadratic quation

geometrically?

For example to solve the equation

x2 + 10 x = 39 he writes

al-Khwarizmi

... a square and 10 roots are equal to 39 units. The question therefore in this type of equation is about as follows: what is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned. Now the roots in the problem before us are 10. Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39 giving 64. Having taken then the square root of this which is 8, subtract from it half the roots, 5 leaving 3. The number three therefore represents one root of this square, which itself, of course is 9. Nine therefore gives the square.

al-Khwarizmi

The geometric proof by completing the square follows. Al-Khwarizmi starts with a square of side x, which therefore represents

x2 (see Figure to follow). To the square we must add 10x and this is done by adding four

rectangles each of breadth 10/4 and length x to the square (see Figure again).

The figure has area x2 + 10 x which is equal to 39. We now complete the square by adding the four little squares each of area 5/2 × 5/2 = 25/4.

Hence the outside square in the Figure has area 4 × 25/4 + 39 = 25 + 39 = 64. The side of the square is therefore 8. But

the side is of length 5/2 + x + 5/2 so x + 5 = 8, giving x = 3.

al-Khwarizmi

al-Khwarizmi and Euclid

These geometrical proofs are a matter of disagreement between experts.

The question, which seems not to have an easy answer, is whether al-Khwarizmi was familiar with Euclid’s Elements.

We know that he could have been, perhaps it is even fair to say "should have been", familiar with Euclid’s work.

al-Khwarizmi and Euclid

In al-Rashid's reign, while al-Khwarizmi was still young, al-Hajjaj had translated Euclid’s Elements into Arabic and al-Hajjaj was one of al-Khwarizmi's colleagues in the House of Wisdom.

This would support the comments of a mathematical historian-

... in his introductory section al-Khwarizmi uses geometrical figures to explain equations, which surely argues for a familiarity with Book II of Euclid’s "Elements".

Al-Khwarizmi

Al-Khwarizmi continues his study of algebra in Hisab al-jabr w'al-muqabala by examining how the laws of arithmetic extend to an arithmetic for his algebraic objects.

For example he shows how to multiply out expressions such as

Al-Khwarizmi

(a+bx)(c+dx)

although again please remember that al-Khwarizmi uses only words to describe his expressions, and no symbols are used.

Rashed (a historian of mathematics) sees a remarkable depth and novelty in these calculations by al-Khwarizmi which appear to us, when examined from a modern perspective, as relatively elementary.

Al-Khwarizmi's

Al-Khwarizmi's concept of algebra can now be grasped with greater precision: it concerns the theory of linear and quadratic equations with a single unknown, and the elementary arithmetic of relative binomials and trinomials. ... The solution had to be general and calculable at the same time and in a mathematical fashion, that is, geometrically founded. ... The restriction of degree, as well as that of the number of unsophisticated terms, is instantly explained. From its true emergence, algebra can be seen as a theory of equations solved by means of radicals, and of algebraic calculations on related expressions...

Al-Khwarizmi

Al-Khwarizmi's algebra is regarded as the foundation and cornerstone of the sciences.

In a sense, al-Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because al-Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus himself is primarily concerned with the theory of numbers and the integer solution to equations

al-Khwarizmi

The next part of al-Khwarizmi's Algebra consists of applications and worked examples.

He then goes on to look at rules for finding the area of figures such as the circle and also finding the volume of solids such as the sphere, cone, and pyramid.

This section on mensuration certainly has more in common with Hindu and Hebrew texts than it does with any Greek work.

al-Khwarizmi

The final part of the book deals with the complicated Islamic rules for inheritance but require little from the earlier algebra beyond solving linear equations.

al-Khwarizmi

Al-Khwarizmi also wrote a treatise on Hindu-Arabic numerals. The Arabic text is lost but a Latin translation, Algoritmi de

numero Indorum in English Al-Khwarizmi on the Hindu Art of Reckoning gave rise to the word algorithm deriving from his name in the title.

Unfortunately the Latin translation (which has been translated into English) is known to be much changed from al-Khwarizmi's original text (of which even the title is unknown).

The work describes the Hindu place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0.

al-Khwarizmi

The first use of zero as a place holder in positional base notation was probably due to al-Khwarizmi in this work.

Methods for arithmetical calculation are given, and a method to find square roots is known to have been in the Arabic original although it is missing from the Latin version.

al-Khwarizmi

The Indian text on which al-Khwarizmi based his treatise was one which had been given to the court in Baghdad around 770 as a gift from an Indian political mission.

There are two versions of al-Khwarizmi's work which he wrote in Arabic but both are lost.

In the tenth century al-Majriti made a critical revision of the shorter version and this was translated into Latin by Adelard.

al-Khwarizmi

There is also a Latin version of the longer version and both these Latin works have survived.

The main topics covered by al-Khwarizmi in the Sindhind zij are calendars; calculating true positions of the sun, moon and planets, tables of sines and tangents; spherical astronomy; astrological tables; parallax and eclipse calculations; and visibility of the moon.

al-Khwarizmi

Although his astronomical work is based on that of the Indians, and most of the values from which he constructed his tables came from Hindu astronomers, al-Khwarizmi must have been influenced by Ptolemy’s work too:-

It is certain that Ptolemy’s tables, in their revision by Theon of Alexandria, were already known to some Islamic astronomers; and it is highly likely that they influenced, directly or through intermediaries, the form in which Al-Khwarizmi's tables were cast.

Al-Khwarizmi

Al-Khwarizmi wrote a major work on geography which give latitudes and longitudes for 2402 localities as a basis for a world map.

The book, which is based on Ptolemy’s Geography, lists with latitudes and longitudes, cities, mountains, seas, islands, geographical regions, and rivers.

The manuscript does include maps which on the whole are more accurate than those of Ptolemy.

Al-Khwarizmi

In particular it is clear that where more local knowledge was available to al-Khwarizmi such as the regions of Islam, Africa and the Far East then his work is considerably more accurate than that of Ptolemy, but for Europe al-Khwarizmi seems to have used Ptolemy’s data.

al-Khwarizmi

A number of minor works were written by al-Khwarizmi on topics such as the astrolabe, on which he wrote two works, on the sundial, and on the Jewish calendar.

He also wrote a political history containing horoscopes of prominent persons.

astrolabe

An astrolabe (Greek: ἁστρολάβον astrolabon 'star-taker') is a historical astronomical instrument used by classical astronomers, navigators, and astrologers.

Its many uses include locating and predicting the positions of the Sun, Moon, planets, and stars; determining local time given local latitude and vice-versa; and surveying.

Arabic mathematics

Recent research paints a new picture of the debt that we owe to Arabic/Islamic mathematics.

Certainly many of the ideas which were previously thought to have been brilliant new conceptions due to European mathematicians of the sixteenth, seventeenth and eighteenth centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier.

Arabic mathematics

In many respects the mathematics studied today is far closer in style to that of the Arabic/Islamic contribution than to that of the Greeks.

There is a widely held view that, after a brilliant period for mathematics when the Greeks laid the foundations for modern mathematics, there was a period of stagnation before the Europeans took over where the Greeks left off at the beginning of the sixteenth century.

Arabic mathematics

The common perception of the period of 1000 years or so between the ancient Greeks and the European Renaissance is that little happened in the world of mathematics except that some Arabic translations of Greek texts were made which preserved the Greek learning so that it was available to the Europeans at the beginning of the sixteenth century.

Arabic mathematics

That such views should be generally held is of no surprise.

Many leading historians of mathematics have contributed to the perception by either omitting any mention of Arabic/Islamic mathematics in the historical development of the subject or with statements such as that made by Duhem (historian)

... Arabic science only reproduced the teachings received from Greek science.

Arabic mathematics

Before proceeding it is worth trying to define the period that we are covering and give an overall description to cover the mathematicians who contributed.

The period covered is easy to describe: it stretches from the end of the eighth century to about the middle of the fifteenth century.

Giving a description to cover the mathematicians who contributed, however, is much harder. There are works on "Islamic mathematics", detailing "Muslim contribution to mathematics".

Arabic mathematics

Other authors try the description "Arabic mathematics“.

However, certainly not all the mathematicians we included were Muslims; some were Jews, some Christians, some of other faiths.

Nor were all these mathematicians Arabs, but for convenience we will call our topic "Arab mathematics".

Arabic mathematics

The regions from which the "Arab mathematicians" came was centred on Iran/Iraq but varied with military conquest during the period.

At its greatest extent it stretched to the west through Turkey and North Africa to include most of Spain, and to the east as far as the borders of China.

Arabic mathematics

The background to the mathematical developments which began in Baghdad around 800 AD is not well understood.

Certainly there was an important influence which came from the Hindu mathematicians whose earlier development of the decimal system and numerals was important.

There began a remarkable period of mathematical progress with al-Khwarizmis work and the translations of Greek texts

Arabic mathematics

This period begins under the Caliph Harun al-Rashid, the fifth Caliph of the Abbasid dynasty, whose reign began in 786.

He encouraged scholarship and the first translations of Greek texts into Arabic, such as Euclid’s Elements by al-Hajjaj, were made during al-Rashid's reign.

Arabic mathematics

The next Caliph, al-Ma'mun, encouraged learning even more strongly than his father al-Rashid, and he set up the House of Wisdom in Baghdad which became the centre for both the work of translating and of of research.

Al-Kindi (born 801) and the three Banu Musa brothers worked there

Arabic mathematics

One should emphasise that the translations into Arabic at this time were made by scientists and mathematicians such as those previously named above, not by language experts ignorant of mathematics, and the need for the translations was stimulated by the most advanced research of the time.

It is important to realise that the translating was not done for its own sake, but was done as part of the current research effort.

Arabic mathematics

Most of the important Greek mathematical texts were translated and list of these exist

Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as "algebraic objects".

It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject.

Arabic mathematics

Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before. One commentary states

Arabic mathematics

Al-Khawarizmi’s successors undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the Euclidean theory of numbers, algebra to geometry, and geometry to algebra. This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose.

Arabic mathematics

Let us follow the development of algebra for a moment and look at Al-Khawarizmi’s successors.

About forty years after Al-Khawarizmi’s is the work of al-Mahani (born 820), who conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.

Abu Kamil (born 850) forms an important link in the development of algebra between Al-Khawarizmi and al-Karaji

Arabic mathematics

Despite not using symbols, but writing powers of x in words, he had begun to understand what we would write in symbols as xn.xm = xm+n.

Let us remark that symbols did not appear in Arabic mathematics until much later.

Ibn al-Banna and al-Qalasadi used symbols in the 15th century and, although we do not know exactly when their use began, we know that symbols were used at least a century before this.

Arabic mathematics

Omar Khayyam (born 1048) gave a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections.

Khayyam also wrote that he hoped to give a full description of the algebraic solution of cubic equations in a later work:-

If the opportunity arises and I can succeed, I shall give all these fourteen forms with all their branches and cases, and how to distinguish whatever is possible or impossible so that a paper, containing elements which are greatly useful in this art will be prepared.

Indian Mathematics

It is without doubt that mathematics today owes a huge debt to the outstanding contributions made by Indian mathematicians over many hundreds of years.

What is quite surprising is that there has been a reluctance to recognise this and one has to conclude that many famous historians of mathematics found what they expected to find, or perhaps even what they hoped to find, rather than to realise what was so clear in front of them.

Indian Mathematics

We shall examine the contributions of Indian mathematics now, but before looking at this contribution in more detail we should say clearly that the "huge debt" is the beautiful number system invented by the Indians on which much of mathematical development has rested.

Laplace put this with great clarity:-

Laplace: Indian mathematics

The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius

Indian mathematics

We shall look briefly at the Indian development of the place-value decimal system of numbers later.

First, however, we go back to the first evidence of mathematics developing in India.

Histories of Indian mathematics used to begin by describing the geometry contained in the Sulbasutras but research into the history of Indian mathematics has shown that the essentials of this geometry were older being contained in the altar constructions described in the Vedic mythology text the Shatapatha Brahmana and the Taittiriya Samhita.

Indian mathematics

Also it has been shown that the study of mathematical astronomy in India goes back to at least the third millennium BC and mathematics and geometry must have existed to support this study in these ancient times.

Indian numerals

Indian numerals

There is no problem in understanding the symbols for 1, 2, and 3. However the symbols for 4, ... , 9 appear to us to have no obvious link to the numbers they represent. There have been quite a number of theories put forward by historians over many years as to the origin of these numerals. Ifrah (historian) lists a number of the hypotheses which have been put forward.

1 The Brahmi numerals came from the Indus valley culture of around 2000 BC. 2 The Brahmi numerals came from Aramaean numerals. 3 The Brahmi numerals came from the Karoshthi alphabet. 4 The Brahmi numerals came from the Brahmi alphabet. 5 The Brahmi numerals came from an earlier alphabetic numeral system, possibly due to

Panini. 6. The Brahmi numerals came from Egypt.

So there is much debate

Brahmi numerals

... the first nine Brahmi numerals constituted the vestiges of an old indigenous numerical notation, where the nine numerals were represented by the corresponding number of vertical lines ... To enable the numerals to be written rapidly, in order to save time, these groups of lines evolved in much the same manner as those of old Egyptian Pharonic numerals. Taking into account the kind of material that was written on in India over the centuries (tree bark or palm leaves) and the limitations of the tools used for writing (calamus or brush), the shape of the numerals became more and more complicated with the numerous ligatures, until the numerals no longer bore any resemblance to the original prototypes.

Brahmi’s numbers

Brahmi numerals

One might have hoped for evidence such as discovering numerals somewhere on this evolutionary path.

However, it would appear that we will never find convincing proof for the origin of the Brahmi numerals.

Brahmi numerals

If we examine the route which led from the Brahmi numerals to our present symbols (and ignore the many other systems which evolved from the Brahmi numerals) then we next come to the Gupta symbols.

The Gupta period is that during which the Gupta dynasty ruled over the Magadha state in North-eastern India, and this was from the early 4th century AD to the late 6th century AD.

The Gupta numerals developed from the Brahmi numerals and were spread over large areas by the Gupta empire as they conquered territory.

Gupta numerals

Nagari numerals

The Gupta numerals evolved into the Nagari numerals, sometimes called the Devanagari numerals.

This form evolved from the Gupta numerals beginning around the 7th century AD and continued to develop from the 11th century onward.

The name literally means the "writing of the gods" and it was the considered the most beautiful of all the forms which evolved. Comments include:-

What we [the Arabs] use for numerals is a selection of the best and most regular figures in India.

Nagari numerals

Indian mathematics

The oldest dated Indian document which contains a number written in the place-value form used today is a legal document dated 346 in the Chhedi calendar which translates to a date in our calendar of 594 AD.

This document is a donation charter of Dadda III of Sankheda in the Bharukachcha region.

The only problem with it is that some historians claim that the date has been added as a later forgery.

Indian mathematics

Although it was not unusual for such charters to be modified at a later date so that the property to which they referred could be claimed by someone who was not the rightful owner, there seems no conceivable reason to forge the date on this document.

Therefore, despite the doubts, we can be fairly sure that this document provides evidence that a place-value system was in use in India by the end of the 6th century.

Origins of zero

Early history By the middle of the 2nd millennium BCE, the Babylonian

mathematics had a sophisticated sexagesimal positional numeral system.

The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals.

By 300 BCE, a punctuation symbol (two slanted wedges) was co-opted as a placeholder in the same Babylonian system.

In a tablet unearthed at Kish (dating from about 700 BCE), the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges.

Origins of zero

The Babylonian placeholder was not a true zero because it was not used alone.

Nor was it used at the end of a number. Thus numbers like 2 and 120 (2×60), 3 and

180 (3×60), 4 and 240 (4×60), looked the same because the larger numbers lacked a final sexagesimal placeholder.

Only context could differentiate them.

Origins of zero

Records show that the ancient Greeks seemed unsure about the status of zero as a number.

They asked themselves, "How can nothing be something?",

Leading to philosophical and, by the Medieval period, religious arguments about the nature and existence of zero and the vacuum.

The paradoxes of Zeno depend in large part on the uncertain interpretation of zero.

Origins of zero

The concept of zero as a number and not merely a symbol for separation is attributed to India where by the 9th century CE practical calculations were carried out using zero, which was treated like any other number, even in case of division.

The Indian scholar Pingala (circa 5th -2nd century BCE) used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), making it similar to Morse code.

He and his contemporary Indian scholars used the Sanskrit word śūnya to refer to zero or void.

History of zero

The Mesoamerican Long Count calendar developed in south-central Mexico and Central America required the use of zero as a place-holder within its vigesimal (base-20) positional numeral system.

Many different glyphs– A glyph is an element of writing

were used as a zero symbol for these Long Count dates, the earliest of which (on Stela 2 at Chiapa de Corzo, Chiapas) has a date of 36 BCE.

History of zero

Since the eight earliest Long Count dates appear outside the Maya homeland it is assumed that the use of zero in the Americas predated the Maya and was possibly the invention of the Olmecs.

The Olmec

The Olmec were an ancient Pre-Columbian civilization living in the tropical lowlands lowlands of south-central Mexico, in what are roughly the modern-day states of Veracruz and Tabasco.

The Olmec flourished during Mesoamerica’s Formative period, dating roughly from 1400 BCE to about 400 BCE. They were the first Mesoamerican civilization and laid many of the foundations for the civilizations that followed.

History of zero

Many of the earliest Long Count dates were found within the Olmec heartland, although the Olmec civilization ended by the 4th century BCE, several centuries before the earliest known Long Count dates.

History of zero

Although zero became an integral part of Maya numerals, it did not influence Old World numeral systems.

The use of a blank on a counting board to represent 0 dated back in India to 4th century BCE.

History of zero

In China counting rods were used for calculation since the 4th century BCE.

Chinese mathematicians understood negative numbers and zero, though they had no symbol for the latter, until the work of the Song Dynasty mathematician Qin Jiushao in 1247 established a symbol for zero in China.

The Nine chapters of the Mathematical Art, which was mainly composed in the 1st century CE, stated "[when subtracting] subtract same signed numbers, add differently signed numbers, subtract a positive number from zero to make a negative number, and subtract a negative number from zero to make a positive

History of zero

By 130 CE, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals.

Because it was used alone, not just as a placeholder, this Hellenistic zero was perhaps the first documented use of a number zero in the Old World.

History of zero

However, the positions were usually limited to the fractional part of a number (called minutes, seconds, thirds, fourths, etc.)—they were not used for the integral part of a number.

In later Byzantine manuscripts of Ptolemy's Syntaxis Mathematica (also known as the Almagest), the Hellenistic zero had morphed into the Greek letter omicron (otherwise meaning 70).

History of zero

Another zero was used in tables alongside Roman numerals by 525 (first known use by Dionsius Exiguus), but as a word, nulla meaning "nothing," not as a symbol.

When division produced zero as a remainder, nihil, also meaning "nothing," was used.

These medieval zeros were used by all future medieval computists (calculators of Easter).

An isolated use of the initial, N, was used in a table of Roman numerals by Bede or a colleague about 725, a zero symbol.

History of zero

In 498 CE, Indian mathematician and astronomer Aryabhata stated that "Sthanam sthanam dasa gunam" or place to place in ten times in value, which may be the origin of the modern decimal-based place value notation.

History of zero

The oldest known text to use a decimal place value system, including a zero, is the Jain text from India entitled the Lokavibhâga, dated 458 CE.

This text uses Sanskrit numeral words for the digits, with words such as the Sanskrit word for void for zero.

History of zero

The first known use of special glyphs for the decimal digits that includes the indubitable appearance of a symbol for the digit zero, a small circle, appears on a stone inscription found in India, dated 876 CE.

There are many documents on copper plates, with the same small o in them, dated back as far as the sixth century CE, but their authenticity may be doubted.

History of zero

The Arabic numerals and the positional number system were introduced to the Islamic civilisation by Al-Khwarizmi.

Al-Khwarizmi's book on arithmetic synthesized Greek and Hindu knowledge and also contained his own fundamental contribution to mathematics and science including an explanation of the use of zero.

It was only centuries later, in the 12th century, that Arabic numeral system was introduced to the Western world through Latin translations of his Arithmetic.